# Computers in Scientific Discovery 5 - Goals

## Background

This workshop is the fifth in a series that started in November 2001 with the Workshop of the DIMACS Working group on Computer-Generated Conjectures from Graph Theoretic and Chemical Databases, and and extends the earlier Discrete Mathematical Chemistry workshop (1998) also held at DIMACS. The events demonstrated the existence of an enthusiastic community of researchers on several continents active at the interfaces between chemistry, computer science and discrete mathematics. Two AMS proceedings volumes resulted (Discrete Mathematical Chemistry).

Computers and Discovery (Workshop 2, Montreal, June 2004) concentrated on the nature of conjecture-making. Nevertheless, several chemical and mathematical software collaborative applications were initiated, such as the "House of Graphs" concept proposed by Pierre Hansen, which aims to align the efforts of groups working on implementing graph algorithms and developing user interfaces for automation of graph handling and conjecture making.

Computers and Scientific Discovery 3 (Ghent, 2006), the first of the series to take place in Europe, concentrated on computer implementation, with direct contact to potential scientific users. To keep developers of mathematical software in touch with their customers, theoretical/mathematical chemistry applications were emphasised, and the scope was extended to mathematical biology and bioinformatics. The next meeting Computers and Scientific Discovery 4 (Shanghai, 2008) again moved to a new continent and carried forward the theme of the connections between the biosciences, mathematics and computer science.

The latest in the series, Computers and Scientific Discovery 5 (Sheffield, 2010) aims to keep up the strong connections and co-operations resulting from the previous workshops and make new contacts between those working on mathematical and software projects and potential users and applications in other fields. A new theme of "Chemoinformatics" will be introduced.

Much of the importance of interdisciplinary meetings comes from their influence in breaking down "language barriers" between fields, and in opening the eyes of researchers in one field to the expertise and challenges available in another. Our workshop format, with formal presentations from invited speakers, contributed talks, but also plenty of time for free discussion between active researchers, should facilitate these aims.

## Chemistry and Chemoinformatics

Since its inception, **graph theory** has had strong connections to **chemistry**, and the revolution in carbon chemistry and
physics that stemmed from the discovery of the fullerenes, nanotubes, graphenes and more exotic forms has given new
prominence in chemistry to arguments and models based on graph theory. Problems associated with graphs and polyhedra
related to the fullerenes have sparked new mathematical investigations, and more and more mathematical concepts,
such as independence number, are finding application in chemistry. Recent work on single-molecule conduction
suggests another explanatory role for graph theory, operating under new boundary conditions.
Mathematical and theoretical chemistry shade into **Chemoinformatics**, which brings together mathematics, computer science
and chemistry to solve chemical problems, often but not exclusively in the area of drug discovery.

A perennial challenge for these multidisciplinary activities is the relation between graph invariants and real-world applications. Many of the classical graph invariants have a starring role in chemistry and chemoinformatics -- qualitative theories of molecular electronic structure depend on properties of the adjacency eigenvalue spectrum, theories of stability and molecular conduction invoke the Kekule count (number of perfect matchings), counting pentagon-pentagon adjacencies gives a first filter for stable fullerene isomers, independence numbers indicate limits to chemical reactivity, and so on. However, there are hundreds ofgraph invariants for which claims for chemical and physical relevance have been made, and some observers think the field is in danger of being choked by its own creativity.

This workshop can help by assessing the practical utility/information content/technical complexity of the more promising invariants, and may both help the users to find a path through the invariant jungle, and perhaps set an agenda for future research.

## Bioinformatics

At least one textbook (of chemoinformatics) maintains that there is no clear distinction between Bioinformatics and Chemoinformatics. Clearly, Bioinformatics deals with intrinsically more complex and delicate networks of systems, and this demands different techniques, even if graph theory and invariants still play major roles in mathematical biology. The workshop will include a contribution on Complexity Science which will explore the use of graph theory for complex systems, and we particularly welcome contributions posing problems in bioinformatics, or discussing the existing and needed mathematical and computer-science tools.

## Conjecture making

Chemical graph theory has been a fertile source for the making of conjectures, and the CSD community has been interested in the study of the conjecture-making process and its potential for automation. The topic was intensively discussed at an earlier workshops, and recent progress will be reviewed at the Sheffield meeting.

## Mathematical Education

Another field of application that has been much discussed at previous workshops is the use of graph-theory and conjecture-making software in teaching, at both high-school and university level. Several groups (e.g. Ghent, Houston, Niš, Montreal) have been working on software that takes different approaches to the teaching of mathematical thinking though the medium of graph theory (see here) and will report progress. Easy-to-use software that allows a user to introduce their own invariants is, of course, also a good candidate for use in the research context.

## Structure Enumeration

Constructing complete lists of mathematical objects has a long tradition in mathematics. The 5 Platonic solids -- that is the complete list of regular polyhedra -- have already been determined by Theaetetus of Athens around 400~B.C. and play the central role in the 13th book of Euclid's "Elements" (around 300 B.C.).

With the arrival of computers the importance of structure enumeration increased. Suddenly it became possible to automatically construct very large lists of structures. These lists were no longer constructed only for their own sake, but sometimes just as tools for further research within and outside of mathematics. Typical applications of this kind are the checking of conjectures on large sets of data, the search for counterexamples and the solution of problems on decisively finite sets (such as e.g. the computation of specific Ramsey numbers).

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